After a few years of operation the electrolytic capacitors on the power supply fail and the monitor can not be powered on or the backlight is not working.

To fix it you must open the monitor and locate the power supply. Visually inspect all capacitors if they are in good condition.

Faulty capacitors are deformed.
Replace any deformed capacitor in the area with the red circle. Replace also the capacitor on the green circle. Its a small capacitor rated at 50 or 63 or 100 volts. This capacitor is responsible for the backlight circuit. Usually this small capacitor fails but does not deform and its hard to tell if its ok or not.